What is the solutions for [tex]2x_1sin\alpha = 0[/tex]?
Obviously one is [tex]x_1=0[/tex], but if [tex]sin\alpha = 0[/tex], then...

Well done. The point is, of course, that [tex]\alpha[/tex] is a number (not one of the variables) so these can be solved like any pair of equations for x1 and x2.
Notice, by the way, that if [tex]sin\alpha= 0[/tex], your first step, dividing by that, would be invalid. You have to look at this case separately.
If [tex]sin\alpha= 0[/tex], then [tex]cos\alpha[/tex] is either 1 or -1.

What do your equations look like if [tex]sin\alpha= 0[/tex] and [tex]cos\alpha= 1[/tex]?

What do your equations look like if [tex]sin\alpha= 0[/tex] and [tex]cos\alpha= -1[/tex]?